Statements and Measures in Statistics

Exploring correct and incorrect statements in statistics

1. Which of the following statements are correct?

1. The class interval in a frequency distribution is the number of data values falling within each class.
2. The largest value in a set of data is 140, and the lowest value is 70. If the resulting frequency distribution is to have five classes of equal width, the class width will be 14.
3. A frequency distribution is a listing of the individual observations arranged in ascending or descending order.
4. An automobile insurance agent believes that company "A" is more reliable than company "B". The scale of measurement that this information represents is the ordinal scale.
5. Chebyshev`s Theorem states that the percentage of observations in a data set that should fall within five standard deviations of their mean is at least 96%.

2. Which of the following statements are incorrect?

1. The value of the standard deviation may be either positive or negative, while the value of the variance will always be positive.
2. Lily has been keeping track of what she spends to eat out. The last week's expenditures for meals eaten out were $5.95, $6.19, $10.91, $7.49, $14.53, and $7.66. The mean amount Lily spends on meals is $8.35.
3. The interquartile range is found by taking the difference between the 1st and 3rd quartiles and dividing that value by 2.
4. Since the sample is always smaller than the population, the sample mean is always smaller than the population mean.
5. A data sample has a mean of 107, a median of 122, and a mode of 134. The distribution of the data is positively skewed.

3. Which of the following statement is incorrect?

1. The value of the mean times the number of observations equals the sum of all of the observations.
2. The median of a set of data would be more representative than the mean of that data when the average of the data values is larger than most of the values.
3. Quartiles divide the values in a data set into four parts of equal size.
4. Professor Hogg graduated from the University of Iowa with a code = 2 while professor Maas graduated from Michigan State University with a code value = 1. The scale of measurement likely represented by this information is ratio.
5. The length of the box in the box-and-whisker plot portrays the interquartile range.


4. Which of the following statement is correct?

Measures of central tendency and dispersion

1. A skewed histogram is one with a long tail extending either to the right or left. The former is called negatively skewed, and the latter is called positively skewed.
2. While Chebyshev`s theorem applies to any distribution, regardless of shape, the empirical rule applies only to distributions that are bell-shaped and symmetrical.
3. The total area of the six bars in a relative frequency histogram for which the width of each bar is five units is 1.
4. When a distribution has values to the right and tails to the left, we say it is skewed positively.
5. The mean is the most frequently used measure of variability.

Part II

Where “Excel” is indicated, please write the Excel function and provide the answer.
The following data represent the ages in years if a sample of 25 employees from a government department:
31     43     56     23     49     42      33     61     44     28
48     38     44     35     40     64      52     42     47     39
53     27     36     35     20
1. Construct a stem and leaf display for the ages. (Manually)

2. Find the median age.

3. Find the lower quartile of the ages.

4. Find the upper quartile of the ages.

5. Find the 60th percentile of the ages.

6. Compute the range and interquartile range of the data.

7. Compute the sample mean age.

8. Compute the sample variance and sample standard deviation.

9. Assuming, that this set of data is a population, compute the population variance and population standard deviation.

10. Construct a box plot for the ages and identify any outliers.

11. Construct a frequency, relative frequency, percent frequency, and cumulative relative frequency distribution for the data, using five classes of 20 -29, 30 - 39, and so on.

12. Construct a relative frequency histogram for the data, using the relative distribution constructed in Question 11.

13. Compare the information regarding skewness conveyed by your box plot constructed in Question 10 with that of the histogram constructed in question 12.

Part III

Please refer to the ‘Superbowl Excel File’, available under the ‘Assignments’ tab on Blackboard.
The Super Bowl has been played in eight states: Arizona (AZ), California (CA), Florida (FL), Georgia (GA), Louisiana (LA), Michigan (MI), Minnesota (MN), and Texas (TX). Data in the Excel file shows the states where the Super Bowls were played and the point margin of victory for the winning teams.
a) Show a frequency distribution and bar chart for the states where the Super Bowls were played.

b) What conclusions can you draw from your summary in part (a)?

c) i) What percentage of Super Bowls were played in the state of Florida?

ii) What percentage of Super Bowls were played in the state of California?